A foot pedal mechanism for a machine is crudely modeled as a...

A foot pedal mechanism for a machine is crudely modeled as a pendulum connected to a spring as illustrated in Figure P1.35. The purpose of the spring is provide a return force for the pedal action. Compute the spring stiffness needed to keep the pendulum at $1^{\circ}$ from the horizontal and then compute the corresponding natural frequency. Assume that the angular deflections are small, such that the spring deflection can be approximated by the arc length, that the pedal may be treated as a point mass and that pendulum rod has negligible mass. The pedal is horizontal when the spring is at its free length. The values in the figure are $m=0.8 \mathrm{~kg}, g=9.8 \mathrm{~m} / \mathrm{s}^{2}, I_{1}=0.2 \mathrm{~m}$ and $l_{2}=0.5 \mathrm{~m}$.

Key Concepts

Arc Length Approximation in Circular Motion

For small angular deflections, the arc length through which a point on a circle moves can be approximated linearly as the product of the radius and the angle in radians. This concept is used to simplify the relationship between angular displacement and linear displacement, which is particularly useful in systems that combine rotational motion with components like springs that typically operate with linear displacements.

Small Angle Approximation

The small angle approximation is a simplification used in rotational dynamics where the sine or tangent of a small angle (in radians) is approximated by the angle itself. This approximation linearizes the equations of motion, making it easier to analyze systems, particularly in the calculation of displacements and forces. It is commonly used in pendulum and spring systems where angular deflections are minimal.

Pendulum Modeling

Pendulum modeling involves representing a mechanical system as a mass attached to a pivot via a rod that can swing under the influence of gravity. In many cases, especially for analytical simplicity, the mass is treated as a point mass and the rod is assumed to be massless. This idealization aids in the derivation of equations of motion and simplifies the calculation of dynamic properties such as oscillation frequencies and equilibrium positions.

Spring Mechanics and Stiffness

Spring mechanics focuses on how springs store and release energy when they are deformed. The stiffness of a spring, defined by Hooke’s law, is a measure of the force required to produce a unit displacement from its equilibrium position. Determining the required spring stiffness is essential in systems where a spring is used to provide a return force or to balance dynamic loads, ensuring that the system maintains a specified equilibrium.

Natural Frequency

The natural frequency of a mechanical system is the frequency at which it oscillates when disturbed from its equilibrium position and allowed to vibrate freely without external driving forces. In linear systems, the natural frequency depends on factors such as mass, stiffness or moment of inertia, and the geometry of the system. It provides important insights into the dynamic behavior and stability of the system.

Transcript

00:03 For the given problem, we give it that a mechanical oscillator when subject to friction forces satisfies the given equation y -double -t -t plus 2y -dash -t plus 5y -t equals to 0.

00:26 We need to prove that identity for the equation, y of t equals e minus t, sine of 2t minus 2kos of 2t.

00:48 We begin by finding y dash of t first.

00:53 So, y dash of t can be found by applying the product rule first.

01:01 We get d by d t of e minus t into sign of two t minus two t minus two cost of two t plus d by d t of sign of two t minus two cost of two t into e raised to the power minus t finding the derivatives of each of these expressions, we get minus e raised to the power minus t, sine of 2t minus 2 cost of 2 t plus d by d t of sine 2 t minus d by d t of cos of 2 t.

02:07 Into e raised to the power minus t.

02:11 In order to find the derivative of each of these underlying expression, we need to apply chain rules separately.

02:19 So to find d by dt of sine of 2t, we assume that the inner function u is equal to 2t, and the outer function v is equal to sign of u.

02:38 Finding the derivative of each of these equations, we get du by dt equals to dv by d .u becomes cost of new.

02:50 Hence, d by dt of sine of 2t can be written as dv by d .u into du by dt by chain rule.

03:04 Thus, substituting the values that we've already found, we get 2 .2.

03:11 Cost of u putting in the value of u we get two costs of two t now finding d by d t of cost two t now the inner function is u equals two t the outer function is cost two t cause of u finding the derivative of each of these expressions we get d u by d t equals 2 d v by d u becomes negative sign of u d by d t cost of 2 t can be written as d v by du by d u into d u by chain rule putting in the values we get 2 minus sign of u simplifying and putting in the value of u we get negative 2, sine of 2t...

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